equal
deleted
inserted
replaced
72 proof (rule measurable_measure_of, simp_all) |
72 proof (rule measurable_measure_of, simp_all) |
73 fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M" |
73 fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M" |
74 using assms[of S] by simp |
74 using assms[of S] by simp |
75 qed |
75 qed |
76 |
76 |
77 lemma borel_measurable_const[measurable (raw)]: |
77 lemma borel_measurable_const: |
78 "(\<lambda>x. c) \<in> borel_measurable M" |
78 "(\<lambda>x. c) \<in> borel_measurable M" |
79 by auto |
79 by auto |
80 |
80 |
81 lemma borel_measurable_indicator: |
81 lemma borel_measurable_indicator: |
82 assumes A: "A \<in> sets M" |
82 assumes A: "A \<in> sets M" |
166 assumes f[measurable]: "f \<in> borel_measurable M" |
166 assumes f[measurable]: "f \<in> borel_measurable M" |
167 assumes g[measurable]: "g \<in> borel_measurable M" |
167 assumes g[measurable]: "g \<in> borel_measurable M" |
168 shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M" |
168 shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M" |
169 and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M" |
169 and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M" |
170 and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
170 and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
171 unfolding eq_iff not_less[symmetric] by measurable+ |
171 unfolding eq_iff not_less[symmetric] |
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172 by measurable |
172 |
173 |
173 subsection "Borel space equals sigma algebras over intervals" |
174 subsection "Borel space equals sigma algebras over intervals" |
174 |
175 |
175 lemma rational_boxes: |
176 lemma rational_boxes: |
176 fixes x :: "'a\<Colon>ordered_euclidean_space" |
177 fixes x :: "'a\<Colon>ordered_euclidean_space" |